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<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">Ah, ok that’s a vote for agda.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">Your ints definition isn’t directly inhabited…..but maybe that doesn’t matter….thanks I’ll struggle on a bit.<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D">(yes my definition is mildly problematic mathematically, I have 3 (coincident) zeros).<o:p></o:p></span></p>
<p class="MsoNormal"><span style="font-size:11.0pt;font-family:"Calibri","sans-serif";color:#1F497D"><o:p> </o:p></span></p>
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<p class="MsoNormal"><b><span lang="EN-US" style="font-size:10.0pt;font-family:"Tahoma","sans-serif"">From:</span></b><span lang="EN-US" style="font-size:10.0pt;font-family:"Tahoma","sans-serif""> Andras Slemmer [mailto:0slemi0@gmail.com]
<br>
<b>Sent:</b> 15 May 2015 10:50 AM<br>
<b>To:</b> Nicholls, Mark<br>
<b>Cc:</b> haskell-cafe@haskell.org<br>
<b>Subject:</b> Re: [Haskell-cafe] Data constructor ‘Minus’ comes from an un-promotable type ‘Ints’<o:p></o:p></span></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal" style="margin-bottom:12.0pt">If you want to stick to your Ints datastructure then you need a stronger dependent type theory that allows you to reason about "double-promoted" types.<o:p></o:p></p>
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<p class="MsoNormal" style="margin-bottom:12.0pt">However in this case you don't really need that complexity:<br>
<br>
data Ints<br>
= Zero<br>
| Minus Nat<br>
| Plus Nat<o:p></o:p></p>
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<p class="MsoNormal">is sufficient enough. Although this definition is probably not the one you want to use... take a look at
<a href="http://www.cse.chalmers.se/~nad/repos/lib/src/Data/Integer.agda">http://www.cse.chalmers.se/~nad/repos/lib/src/Data/Integer.agda</a> for hints<o:p></o:p></p>
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<p class="MsoNormal"><o:p> </o:p></p>
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<p class="MsoNormal">On 14 May 2015 at 18:36, Nicholls, Mark <<a href="mailto:nicholls.mark@vimn.com" target="_blank">nicholls.mark@vimn.com</a>> wrote:<o:p></o:p></p>
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<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Hello,<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">I clearly don’t really know what I’m doing…but at least I know it….<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Here we defined the Naturals…and then attempt to construct the Integers….<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE DataKinds #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE ExplicitForAll #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE FlexibleContexts #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE FlexibleInstances #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE GADTs #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE MultiParamTypeClasses #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE PolyKinds #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE StandaloneDeriving #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE TypeFamilies #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE TypeOperators #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE UndecidableInstances #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> {-# LANGUAGE ScopedTypeVariables #-}<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> import Prelude hiding (head, tail, (++), (+), replicate)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> import qualified Prelude as P<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">naturals<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> data Nat where<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> Z :: Nat<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> S :: Nat -> Nat<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">I can now define + and * and prove things about them…1 * x == x etc….nice..but lets put that on one side.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Borrow bits and bobs from singletons<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> data family Sing (a :: k)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Borrow bits and bobs from singletons..i.e. the isomorphic values…my proofs in nat now map to SNat…double nice.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> type SNat = (Sing :: Nat -> *)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Create the integers by following my nose……(the integers are the equivalence class of pairs of naturals….)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">i.e. we have “positive” or “negative” or “zero”…<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> data Ints (a :: Nat) (b :: Nat) where<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> Minus :: SNat a -> Ints 'Z a<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> Zero :: Ints 'Z 'Z<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">> Plus :: SNat a -> Ints a 'Z<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">>
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Ok….this works as a set of values….but….<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">I can’t prove anything about these because the data constructors for my integers aren’t “promotable”…..so I cant do the same trick I did with Nat.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">“:k Zero” …..<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto;text-autospace:none">
<span style="font-family:Consolas;color:#AF0000"> Data constructor ‘Zero’ comes from an un-promotable type ‘Ints’</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"><span style="font-family:Consolas;color:#AF0000"> In a type in a GHCi command: Zero</span><o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">I’ve tried rejigging this in various futile and ignorant manners….but the bottom line is its…”un-promotable”…..look at the docs, and it says something vague about GADT’s…but that
isnt the problem directly.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Is there a nifty way around this log jam?
<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">(I can start proving things about ‘(Nat,Nat)….but this will soon become a bit clumsy….)<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">Or do I just stop here and wrestle with getting agda installed and lose another few months in the agda-café (which appears to be a very nice place somewhere in finland….hmmmm…I
think that’s something different).<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">(I’ve already allegedly descended into cabal hell following my nose with an agda install…<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">setup.exe: The program cpphs version >=1.18.6 && <1.19 is required but the<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">version found at C:\Users\nichom\AppData\Roaming\cabal\bin\cpphs.exe is<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">version 1.19).<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto">computer science would be good, if it wasn’t for the computers.<o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
<p class="MsoNormal" style="mso-margin-top-alt:auto;mso-margin-bottom-alt:auto"> <o:p></o:p></p>
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